More in general, the dependance of \(\mathfrak {a}_{\varepsilon }\)  and \(\mathfrak {b}_{\varepsilon }\)  on \(\varepsilon \)  only involves elementary operations but is somewhat unpleasant to write down explicitly. The proof follows directly from elementary formulas in Section 5 of [1]}, which provide explicit upper bounds in terms of volume and injectivity radius. In the case of coexact 1-forms there is an extra constant term \(C_{\varepsilon }\)  (depending only on \(\varepsilon \) ) which we convert into the desired form by
\(C_{\varepsilon }=\frac{C_{\varepsilon }}{\mathrm {vol(Y)}}\cdot \mathrm {vol(Y)}\le \frac{C_{\varepsilon }}{0.94}\cdot \mathrm {vol(Y)},\) 
